Kurt Gödel

Kurt Gödel's father was Rudolf Gödel whose family were from Vienna. Rudolf did not take his academic studies far as a young man, but had done well for himself becoming managing director and part owner of a major textile firm in Brünn. Kurt's mother, Marianne Handschuh, was from the Rhineland and the daughter of Gustav Handschuh who was also involved with textiles in Brünn. Rudolf was 14 years older than Marianne who, unlike Rudolf, had a literary education and had undertaken part of her school studies in France. Rudolf and Marianne Gödel had two children, both boys. The elder they named Rudolf after his father, and the younger was Kurt.

Kurt had quite a happy childhood. He was very devoted to his mother but seemed rather timid and troubled when his mother was not in the home. He had rheumatic fever when he was six years old, but after he recovered life went on much as before. However, when he was eight years old be began to read medical books about the illness he had suffered from, and learnt that a weak heart was a possible complication. Although there is no evidence that he did have a weak heart, Kurt became convinced that he did, and concern for his health became an everyday worry for him.

Kurt attended school in Brünn, completing his school studies in 1923. His brother Rudolf said:-

Even in High School my brother was somewhat more one-sided than me and to the astonishment of his teachers and fellow pupils had mastered university mathematics by his final Gymnasium years. ... Mathematics and languages ranked well above literature and history. At the time it was rumoured that in the whole of his time at High School not only was his work in Latin always given the top marks but that he had made not a single grammatical error.

Gödel entered the University of Vienna in 1923 still without having made a definite decision whether he wanted to specialise in mathematics or theoretical physics. He was taught by Furtwängler, Hahn, Wirtinger, Menger, Helly and others. The lectures by Furtwängler made the most impact on Gödel and because of them he decided to take mathematics as his main subject. There were two reasons: Furtwängler was an outstanding mathematician and teacher, but in addition he was paralysed from the neck down so lectured from a wheel chair with an assistant who wrote on the board. This would make a big impact on any student, but on Gödel who was very conscious of his own health, it had a major influence. As an undergraduate Gödel took part in a seminar run by Schlick which studied Russell's book Introduction to mathematical philosophy. Olga Taussky-Todd, a fellow student of Gödel's, wrote:-

It became slowly obvious that he would stick with logic, that he was to be Hahn's student and not Schlick's, that he was incredibly talented. His help was much in demand.

He completed his doctoral dissertation under Hahn's supervision in 1929 submitting a thesis proving the completeness of the first order functional calculus. He became a member of the faculty of the University of Vienna in 1930, where he belonged to the school of logical positivism until 1938. Gödel's father died in 1929 and, having had a successful business, the family were left financially secure. After the death of her husband, Gödel's mother purchased a large flat in Vienna and both her sons lived in it with her. By this time Gödel's older brother was a successful radiologist. We mentioned above that Gödel's mother had a literary education and she was now able to enjoy the culture of Vienna, particularly the theatre accompanied by Rudolf and Kurt.

Gödel is best known for his proof of "Gödel's Incompleteness Theorems". In 1931 he published these results in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. This ended a hundred years of attempts to establish axioms which would put the whole of mathematics on an axiomatic basis. One major attempt had been by Bertrand Russell with Principia Mathematica (1910-13). Another was Hilbert's formalism which was dealt a severe blow by Gödel's results. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisaged by Hilbert. Gödel's results were a landmark in 20th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions.

Gödel met Zermelo in Bad Elster in 1931. Olga Taussky-Todd, who was at the same meeting, wrote:-

The trouble with Zermelo was that he felt he had already achieved Gödel's most admired result himself. Scholz seemed to think that this was in fact the case, but he had not announced it and perhaps would never have done so. ... The peaceful meeting between Zermelo and Gödel at Bad Elster was not the start of a scientific friendship between two logicians.

Submitting his paper on incompleteness to the University of Vienna for his habilitation, this was accepted by Hahn on 1 December 1932. Gödel became a Privatdozent at the University of Vienna in March 1933.

Now 1933 was the year that Hitler came to power. At first this had no effect on Gödel's life in Vienna; he had little interest in politics. In 1934 Gödel gave a series of lectures at Princeton entitled On undecidable propositions of formal mathematical systems. At Veblen's suggestion Kleene, who had just completed his Ph.D. thesis at Princeton, took notes of these lectures which have been subsequently published. However, Gödel suffered a nervous breakdown as he arrived back in Europe and telephoned his brother Rudolf from Paris to say he was ill. He was treated by a psychiatrist and spent several months in a sanatorium recovering from depression.

Despite the health problems, Gödel's research was progressing well and he proved important results on the consistency of the axiom of choice with the other axioms of set theory in 1935. However after Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by a National Socialist student in 1936, Gödel was much affected and had another breakdown. His brother Rudolf wrote:-

This event was surely the reason why my brother went through a severe nervous crisis for some time, which was of course of great concern, above all for my mother. Soon after his recovery he received the first call to a Guest Professorship in the USA.

He visited Göttingen in the summer of 1938, lecturing there on his set theory research. He returned to Vienna and married Adele Porkert in the autumn of 1938. In fact he had met her in 1927 in Der Nachtfalter night club in Vienna. She was six years older than Gödel and had been married before and both his parents, but particularly his father, objected to the idea that they marry. She was not the first girl that Gödel's parents had objected to, the first he had met around the time he went to university was ten years older than him.

In March 1938 Austria had became part of Germany but Gödel was not much interested and carried on his life much as normal. He visited Princeton for the second time, spending the first term of session 1938-39 at the Institute for Advanced Study. The second term of that academic year he gave a beautiful lecture course at Notre Dame. Most who held the title of privatdozent in Austria became paid lecturers after the country became part of Germany but Gödel did not and his application made on 25 September 1939 was given an unenthusiastic response. It seems that he was thought to be Jewish, but in fact this was entirely wrong, although he did have many Jewish friends. Others also mistook him for a Jew, and he was once attacked by a gang of youths, believing him to be a Jew, while out walking with his wife in Vienna.

When the war started Gödel feared that he might be conscripted into the German army. Of course he was also convinced that he was in far too poor health to serve in the army, but if he could be mistaken for a Jew he might be mistaken for a healthy man. He was not prepared to risk this, and after lengthy negotiation to obtain a U.S. visa he was fortunate to be able to return to the United States, although he had to travel via Russia and Japan to do so. His wife accompanied him.

In 1940 Gödel arrived in the United States, becoming a U.S. citizen in 1948 (in fact he believed he had found an inconsistency in the United States Constitution, but the judge had more sense than to listen during his interview!). He was an ordinary member of the Institute for Advanced Study from 1940 to 1946 (holding year long appointments which were renewed every year), then he was a permanent member until 1953. He held a chair at Princeton from 1953 until his death, holding a contract which explicitly stated that he had no lecturing duties. One of Gödel's closest friends at Princeton was Einstein. They each had a high regard for the other and they spoke frequently. It is unclear how much Einstein influenced Gödel to work on relativity, but he did indeed contribute to that subject.

He received the Einstein Award in 1951, and National Medal of Science in 1974. He was a member of the National Academy of Sciences of the United States, a fellow of the Royal Society, a member of the Institute of France, a fellow of the Royal Academy and an Honorary Member of the London Mathematical Society. However, it says much about his feelings towards Austria that he refused membership of the Academy of Sciences in Vienna, then later when he was elected to honorary membership he again refused the honour. He also refused to accept the highest National Medal for scientific and artistic achievement that Austria offered him. He certainly felt bitter at his own treatment but equally so about that of his family.

Gödel's mother had left Vienna before he did, for in 1937 she returned to her villa in Brno where she was openly critical of the National Socialist regime. Gödel's brother Rudolf had remained in Vienna but by 1944 both expected German defeat, and Rudolf's mother joined him in Vienna. In terms of the treaty negotiated after the war between the Austrians and the Czechs, she received one tenth of the value for her villa in Brno. It was an injustice which infuriated Gödel; in fact he always took such injustices as personal even although large numbers suffered in the same way.

After settling in the United States, Gödel again produced work of the greatest importance. His masterpiece Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940) is a classic of modern mathematics. In this he proved that if an axiomatic system of set theory of the type proposed by Russell and Whitehead in Principia Mathematica is consistent, then it will remain so when the axiom of choice and the generalized continuum-hypothesis are added to the system. This did not prove that these axioms were independent of the other axioms of set theory, but when this was finally established by Cohen in 1963 he built on these ideas of Gödel.

Concerns with his health became increasingly worrying to Gödel as the years went by. Rudolf, Gödel's brother, was a medical doctor so the medical details given by him in the following will be accurate. He wrote:-

My brother had a very individual and fixed opinion about everything and could hardly be convinced otherwise. Unfortunately he believed all his life that he was always right not only in mathematics but also in medicine, so he was a very difficult patient for doctors. After severe bleeding from a duodenal ulcer ... for the rest of his life he kept to an extremely strict (over strict?) diet which caused him slowly to lose weight.

Adele, Gödel's wife, was a great support to him and she did much to ease the tensions which troubled him. However she herself began to suffer health problems, having two strokes and a major operation. Towards the end of his life Gödel became convinced that he was being poisoned and, refusing to eat to avoid being poisoned, essentially starved himself to death [3]:-

A slight person and very fastidious, Gödel was generally worried about his health and did not travel or lecture widely in later years. He had no doctoral students, but through correspondence and personal contact with the constant succession of visitors to Princeton, many people benefited from his extremely quick and incisive mind. Friend to Einstein, von Neumann and Morgenstern, he particularly enjoyed philosophical discussion.

He died [18]:-

... sitting in a chair in his hospital room at Princeton, in the afternoon of 14 January 1978.

It would be fair to say that Gödel's ideas have changed the course of mathematics [3]:-

... it seems clear that the fruitfulness of his ideas will continue to stimulate new work. Few mathematicians are granted this kind of immortality.

Article by:J J O'Connor and E F Robertson

October 2003

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Godel.html]